 
Summary: A DIXMIERMOEGLIN EQUIVALENCE FOR
POISSON ALGEBRAS WITH TORUS ACTIONS
K. R. Goodearl
Abstract. A Poisson analog of the DixmierMoeglin equivalence is established for any affine
Poisson algebra R on which an algebraic torus H acts rationally, by Poisson automorphisms,
such that R has only finitely many prime Poisson Hstable ideals. In this setting, an additional
characterization of the Poisson primitive ideals of R is obtained they are precisely the prime
Poisson ideals maximal in their Hstrata (where two prime Poisson ideals are in the same
Hstratum if the intersections of their Horbits coincide). Further, the Zariski topology on
the space of Poisson primitive ideals of R agrees with the quotient topology induced by the
natural surjection from the maximal ideal space of R onto the Poisson primitive ideal space.
These theorems apply to many Poisson algebras arising from quantum groups.
The full structure of a Poisson algebra is not necessary for the results of this paper, which
are developed in the setting of a commutative algebra equipped with a set of derivations.
Introduction
Motivated by existing and conjectured roles of Poisson structures in the theory of quan
tum groups, we address some problems in the ideal theory of Poisson algebras. Recall,
for example, that Hodges and Levasseur [15, 16] and Joseph [18] have constructed bi
jections between the primitive ideal space of the quantized coordinate ring Oq(G) of a
semisimple Lie group G and the set of symplectic leaves in G corresponding to a Poisson
